两平行导线间轴向运动载流梁的非线性主共振
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摘要
研究轴向运动载流梁在两平行导线产生磁场中的主共振问题。给出两平行导线间载流梁处磁感应强度及所受电磁力表达式,推得轴向运动载流梁的横向振动微分方程。应用伽辽金积分法,得到轴向运动梁无量纲化的非线性振动微分方程。采用多尺度法进行求解,得到系统关于前两阶模态非线性方程的近似解析解以及主共振幅频响应方程。通过算例,得到了轴向运动载流梁共振幅值随调谐参数、载流电流密度、导线电流和位置的变化关系曲线图。结果表明,各相关物理和几何参数的改变对系统共振特征有较大影响,且非线性振动特征较为明显。
The magneto-elastic primary resonance of an axially moving current-carrying beam is investigated, where the beam move between two parallel and infinite long straight current-carrying wires. According to the principles of an electromagnetic field, the expressions of the electromagnetic force loading on the current-carrying beam is developed. Based on the Hamiltonian principle, the transverse vibration control equations of an axially moving current-carrying beam is derived. The non-dimensional nonlinear differential equation of an axially moving beam is obtained by means of Galerkin method. The approximate analytical solution of first two-order modal nonlinear equation and the primary resonance-amplitude-frequency response equation are derived by means of the multiple scales method. Through computational examples, the resonance amplitude of the axially moving current-carrying beam varying with frequency parameters, current-carrying density, current of wire, and the position variation relationship are obtained. The results show that there is an obvious nonlinear behavior and a great influence on the resonance characteristics of the system when the relevant physical and geometric parameters are changed.
The magneto-elastic primary resonance of an axially moving current-carrying beam is investigated, where the beam move between two parallel and infinite long straight current-carrying wires. According to the principles of an electromagnetic field, the expressions of the electromagnetic force loading on the current-carrying beam is developed. Based on the Hamiltonian principle, the transverse vibration control equations of an axially moving current-carrying beam is derived. The non-dimensional nonlinear differential equation of an axially moving beam is obtained by means of Galerkin method. The approximate analytical solution of first two-order modal nonlinear equation and the primary resonance-amplitude-frequency response equation are derived by means of the multiple scales method. Through computational examples, the resonance amplitude of the axially moving current-carrying beam varying with frequency parameters, current-carrying density, current of wire, and the position variation relationship are obtained. The results show that there is an obvious nonlinear behavior and a great influence on the resonance characteristics of the system when the relevant physical and geometric parameters are changed.
引文
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[20]OzkayaE,PakdemirliM.Vibrationsofanaxially acceleratingbeamwithsmallflexuralstiffness[J].Journal of Sound and Vibration, 2000, 234(3):521-535.
[21]DingHu,ChenLiqun.Galerkinmethodsfornatural frequenciesofhigh-speedaxiallymovingbeams[J].JournalofSoundandVibration,2010,329(17):3484-3494.
[22]刘金堂,杨晓东,张宇飞.轴向运动大挠度板的非线性动力学行为[J].工程力学, 2011, 28(10):58-64.LiuJintang,YangXiaodong,ZhangYufei.Nonlinear dynamicalbehaviorsofaxiallymovinglargedeflection plates[J]. Engineering Mechanics, 2011, 28(10):58-64.(in Chinese)
[23]YangXiaodong,ZhangWei.Nonlineardynamicsof axiallymovingbeamwithcoupledlongitudinaltransversalvibrations[J].NonlinearDynamics,2014,78(78):2547-2556.
[2]胡宇达.轴向运动导电薄板磁弹性耦合动力学理论模型[J].固体力学学报, 2013, 34(4):417-425.HuYuda.Magneto-elasticcoupleddynamicstheoretical modelofaxiallymovingcurrent-conductingthinplate[J].ChineseJournalofSolidMechanics,2013,34(4):417-425.(in Chinese)
[3]RiedelCH,TanCA.Coupledforcedresponseofan axiallymovingstripwithinternalresonance[J].InternationalJournalofNon-LinearMechanics,2002,37(1):101―116.
[4]黄建亮,陈树辉.不同运动速度下轴向运动梁的非线性振动研究[J].中山大学学报, 2008, 47(2):1-4.HuangJianliang,ChenShuhui.Studyonnonlinear vibrationofaxiallymovingbeamswithvarying velocities[J].JournalofSun Yat-senUniversity,2008,47(2):1-4.(in Chinese)
[5]李成,随岁寒,杨昌锦.受初应力作用的轴向运动功能梯度梁的动力学分析[J].工程力学,2015,32(10):226-232.Li Cheng, Sui Suihan, Yang Changjin. Dynamic analysis of axially moving functionally graded beams subjected to initialstress[J].EngineeringMechanics,2015,32(10):226-232.(in Chinese)
[6]李成,姚林泉.轴向运动超薄梁的非局部动力学分析[J].工程力学, 2013, 30(4):367-372.Li Cheng, Yao Linquan. Nonlocal dynamical analysis on axiallytravellingultra-thinbeams[J].Engineering Mechanics, 2013, 30(4):367-372.(in Chinese)
[7]胡宇达,孙建涛,张金志.横向磁场中轴向变速运动矩形板的参数振动[J].工程力学,2013,30(9):299-304.Hu Yuda, Sun Jiantao, Zhang Jinzhi. Parametric vibration ofaxiallyacceleratingrectangularplateintransverse magneticfield[J].EngineeringMechanics,2013,30(9):299-304.(in Chinese)
[8]陈强,杨国来,王晓锋.轴向变速运动弯曲梁的固有频率分析[J].工程力学, 2015, 32(2):37-44.ChenQiang,YangGuolai,WangXiaofeng.Analysisof thenaturalfrequencyofanaxiallymovingbeamwith non-uniformvelocity[J].EngineeringMechanics,2015,32(2):37-44.(in Chinese)
[9]XueChunxia,PanE,HanQike,etal.Non-linear principal resonance of an orthotropic and magneto-elastic rectangular plate[J]. International Journal of Non-Linear Mechanics, 2011, 46(5):703-710.
[10]LiXiaojun,ChenLiqun.Transversevibrationand stabilityofanaxiallymovingbeamwithpinnedand fixedends[J].JournalofMechanicalStrength,2006,28(5):654―657.
[11]PakdemirliM,BoyacmH.Non-linearvibrationsand stability of an axially moving beam with time-dependent velocity[J].InternationalJournalofNon-Linear Mechanics, 2001, 36(1):107-115.
[12]PratiherB.Non-linearresponseofamagneto-elastic translatingbeamwithprismaticjointforhigher resonanceconditions[J].InternationalJournalof Non-linear Mechanics, 2011, 46(5):685-692.
[13]PellicanoF,ZirilliF.Boundarylayersandnon-linear vibrations in an axially moving beam[J]. British Journal of Surgery, 1998, 75(33):691-711.
[14]ChenLiqun,YangXiaodong.Steady-stateresponseof axiallymovingviscoelasticbeamswithpulsatingspeed:comparisonoftwononlinearmodels[J].International Journal of Solids&Structures, 2005, 42(1):37-50.
[15]Wu Guanyuan. The analysis of dynamic instability on the largeamplitudevibrationsofabeamwithtransverse magneticfieldsandthermalloads[J].JournalofSound and Vibration, 2007, 302(1/2):167―177.
[16]GhayeshMH,AmabiliM.Nonlinearvibrationsand stability of an axially moving Timoshenko beam with an intermediatespringsupport[J].Mechanism&Machine Theory, 2013, 67(67):1-16.
[17]GhayeshMH,KafiabadHA,TylerR.Suband super-criticalnonlineardynamicsofaharmonically excited axially moving beam[J]. International Journal of Solids and Structures, 2012, 49(1):227-243.
[18]HuYuda,ZhangJinzhi.Principalparametricresonance of axially accelerating rectangular thin plate in magnetic field[J].AppliedMathematicsandMechanics,2013,34(11):1405-1420.
[19]HuYuda,HuPeng,ZhangJinzhi.Stronglynonlinear subharmonicresonanceandchaoticmotionofaxially movingthinplateinmagneticfield[J].Journalof ComputationalandNonlinearDynamics,2015,10(2):021010-1-021010-12.
[20]OzkayaE,PakdemirliM.Vibrationsofanaxially acceleratingbeamwithsmallflexuralstiffness[J].Journal of Sound and Vibration, 2000, 234(3):521-535.
[21]DingHu,ChenLiqun.Galerkinmethodsfornatural frequenciesofhigh-speedaxiallymovingbeams[J].JournalofSoundandVibration,2010,329(17):3484-3494.
[22]刘金堂,杨晓东,张宇飞.轴向运动大挠度板的非线性动力学行为[J].工程力学, 2011, 28(10):58-64.LiuJintang,YangXiaodong,ZhangYufei.Nonlinear dynamicalbehaviorsofaxiallymovinglargedeflection plates[J]. Engineering Mechanics, 2011, 28(10):58-64.(in Chinese)
[23]YangXiaodong,ZhangWei.Nonlineardynamicsof axiallymovingbeamwithcoupledlongitudinaltransversalvibrations[J].NonlinearDynamics,2014,78(78):2547-2556.